Definitions

Two semgents (or rays) are parallel if the lines that contain them are parallel.

ParallelNot skew

Not ParallelNot skew

Not ParallelSkew

Note that the middle example shows a hole in our definitions. There's a possible arrangement of the lines that is neither parallel nor skew. Geometry books
rarely give this case an official definition but we usually refer to them as "intersecting".

You should also note that the definition of skew lines implies that they don't intersect because two intersecting lines always lie in the same plane.